Many image processing noise reduction algorithms can be classified as non-linear spatial filters. Often these algorithms involve using the pixel values in a small local neighborhood surrounding the pixel of interest combined with some form of non-linear weighting and/or statistical conditions applied to the pixels in the neighborhood to derive a noise free estimate of the pixel of interest. The small local neighborhood is usually centered on the pixel of interest. For this class of noise reduction algorithms the filter size is fixed, meaning that all image pixels are processed with the same size local neighborhood. The most common shape to the local neighborhood is a rectangular region centered about the pixel of interest. Such a region can be characterized by a width and height. Usually the width and height dimensions are chosen to be symmetric.
An example of a fixed size rectangular region noise reduction algorithm is the Sigma Filter, described by Jong-Sen Lee in the journal article “Digital Image Smoothing and the Sigma Filter”, Computer Vision, Graphics, and Image Processing, Vol. 24, 1983, pp. 255-269. This is a noise reduction filter that uses a non-linear pixel averaging technique sampled from a rectangular window about the center pixel. Pixels in the local neighborhood are either included or excluded from the numerical average on the basis of the difference between the pixel and the center pixel. Mathematically, the Sigma Filter can be represented asqmn=Σij aij pij/Σijaij andaij=1 if |pij−pmn|>=εaij=0 if |pij−pmn|>εwhere pij represents the pixels in the local surround about the center pixel pmn, qmn represents the noise cleaned pixel, and ε represents a numerical constant usually set to two times the expected noise standard deviation. The local pixels are sampled from a rectangular region centered about the pixel of interest.
The Sigma Filter was designed for image processing applications for which the dominant noise source is Gaussian additive noise. Signal dependent noise sources can easily be incorporated by making the e parameter a function of the signal strength. However, for both signal independent and signal dependent noise cases the expected noise standard deviation must be known to obtain optimal results. The Sigma Filter performs well on highly structured areas due to the fact that most of the image pixels in the local neighborhood are excluded from the averaging process. This leaves high signal strength regions nearly unaltered. The filter also works well in large uniform areas devoid of image signal structure due to the fact that most of the local pixels are included in the averaging process. For these regions, the Sigma Filter behaves as a low pass spatial filter with a rectangular shape. This low-pass spatial filter shape does not filter very low spatial frequency components of the noise. The resulting noise reduced images can have a blotchy or mottled appearance in otherwise large uniform areas.
Regions in images characterized by low amplitude signal modulation, or low signal strength, are not served well by the Sigma Filter. For these regions, most of the local pixel values are included in the averaging process thus resulting in a loss of signal modulation. Setting the threshold of the filter to a lower value does reduce the loss of signal, however, the noise is left mostly the same.
Another example of a fixed size non-linear noise filter was reported by Arce and McLoughlin in the journal article “Theoretical Analysis of the Max/Median Filter”, IEEE Transactions Acoustics, Speech and Signal Processing, ASSP-35, No. 1, January 1987, pp. 60-69, they named the Max/Median Filter. This filter separated the local surround region into four overlapping regions horizontal, vertical, and two diagonal pixels with each region containing the center pixel. A pixel estimate was calculated for each region separately by applying and taking the statistical median pixel value sampled from the regions' pixel values. Of these four pixel estimates, the maximum valued estimate was chosen as the noise cleaned pixel. Mathematically the Max/Median Filter can be represented asqij=maximum of {Z1, Z2, Z3, Z4}                Z1=median of {pi,j−w, . . . pij, . . . , pij+w}        Z2=median of {pi−w,j, . . . pi,j, . . . , pi+w,j}        Z3=median of {pi+w,j−w, . . . pi,j, . . . ,pi−w,j+w}        Z4=median of {pi−w,j−w, pi,j, . . . ,pi+w,j+w}Where qij represents the noise cleaned pixel, Z1, Z2, Z3, and Z4 represent the four pixel estimates, and pij represents the local pixel values. The Max/Median Filter also reduces the noise present while preserving edges. For Gaussian additive noise, the statistical median value does not reduce the noise by as great a factor as numerical averaging. However, this filter does work well on non-Gaussian additive noise such as spurious noise.        
Noise is most visible and objectionable in images containing areas with little signal structure, e.g. blue sky regions with little or no clouds. The Sigma filter can produce a blotchy, or mottled, effect when applied to image regions characterized by low signal content. This is largely due to the rectangular geometric sampling of local pixels strategy. The radial region sampling strategy employed by the Max/Median Filter produces noise reduced images with less objectionable artifacts in image regions characterized by low signal content. For images with high noise content, the artifacts produced by radial region sampling strategy have a structured appearance.
U.S. Pat. No. 5,671,264, issued Sep. 23, 1997 to Florent et al., entitled “Method for the Spatial Filtering of the Noise in a Digital Image, and Device for Carrying Out the Method”, describes a variation of the Sigma Filter and Max/Median Filter. This algorithm borrows the technique of radial spatial sampling and multiple pixel estimates from the Max/Median Filter. However, the algorithm expands the number of radial line segment to include configurations with more than four segments. The algorithm uses combinations of Sigma and Median filters to form the individual region pixel estimates. These pixel estimates derived from the N regions are then combined by numerical averaging or taking the statistical median value to form the noise cleaned pixel value. A key component of this algorithm is the randomization of one of the three essential region parameters: length, orientation, and number of regions. The randomization of the filter parameters is performed on a pixel to pixel basis thus changing the inherent characteristics with pixel location. It is claimed that the randomization feature reduces the induced structured artifacts produced by the radial region geometry sampling method. The imaging application cited in U.S. Pat. No. 5,671,264 is medial x-ray imagery. This type of imagery is typically characterized by high noise content or a low signal-to-noise ratio. The structured artifacts introduced by the noise reduction algorithm are worse for low signal-to-noise ratio images.
Fixed rectangular local surround noise reduction can produce spatial artifacts. Algorithms employing small filter sizes take less computation time and preserve desirable low amplitude modulation signals but are also less affective at removing noise in unstructured regions. Algorithms employing large filter sizes take more computation time and are more effective at removing noise in unstructured regions but also destroy desirable low amplitude modulation signals and leave blotchy low spatial frequency noise artifacts. Radial region based noise reduction algorithms are effective at removing noise in unstructured regions but can produce unwanted structured patterns in the noise cleaned images. These unwanted spatial artifacts are highly dependent on the type of imagery processed, and in particular, on the signal-noise-ratio of the imagery. Varying the size, orientation, or number of radial regions randomly can reduce the objectionability of the unwanted structured patterns for low signal-to-noise ratio imagery but requires more computation time and complexity to switch filter patterns. What is needed is a noise reduction algorithm which uses a radial pattern of local pixels to reduce the structured artifacts without the computational complexity of randomly switching patterns.